Chapter 11 · Book pp. 272–282 · August 9, 2026 edition
Non-Log-Concave Sampling
Use Fisher information to formulate and analyze approximate stationarity for non-log-concave targets, then compare upper and lower bounds.
Begin with 11.1 Open this chapter in the canonical August 9 source ↗Chapter route
This chapter develops Fisher information, first-order stationarity, nonconvex Langevin dynamics, dissipativity. Its main destination is to connect the definitions below to the results that later chapters consume.
Core definitions
- Relative Fisher information measures the squared score mismatch against the target.
- Approximate first-order stationarity is formulated through small Fisher information rather than small sampling distance.
- Dissipativity controls the drift outside a bounded region without global convexity.
Main results
- Small Fisher information provides a meaningful stationarity certificate for non-log-concave targets.
- Langevin and related algorithms admit finite-time Fisher-information bounds under stated regularity and moment assumptions.
- Fisher-information control yields consequences for optimization and sampling observables.
- Lower bounds delimit the attainable stationarity rates.
Contents
Why is this chapter route valid?
Analytic contracts
- Relative Fisher information needs an absolutely continuous law and a chosen score representative.
- Entropy dissipation must be justified for the actual process and function domain, not only calculated formally.
- Randomized-time output requires measurability of the time-indexed law and Fisher-information integrand.
- Finite-time discretization and score-error bounds must specify the law under which every squared error is integrated.
- Small Fisher information is a stationarity certificate, not automatically small total variation or rapid multimodal mixing.
Open boundaries
- Relative-score representative and Fisher-information API
- Rigorous entropy-dissipation/domain packet
- Langevin discretization-to-Fisher-information chain
- Book-specific applications and lower bounds
View Lean formalization
These mappings are evidence links, not a claim that the entire chapter is formalized.
Relative Fisher information is used as a first-order stationarity measure for non-log-concave sampling, and entropy dissipation supplies an averaged finite-time bound.